ELASTIC WAVE PROPAGATION
IN HOMOGENEOUS TRANSVERSELY ISOTROPIC MEDIUM
WITH SYMMETRY AXIS PARALLEL TO THE FREE SURFACE
M.A.,
by
Chi-fung Wang
B.S., National Taiwan University
(1966)
State University of New York at Stoney Brook
(1970)
SUBMITTED IN
PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY
September, 1973
Signature of Author.
Depar+-m"n+- of
'arth(-andUlanetarv_
.-- nces
/7J
Certified by..
isis Supervisor
b/ i3 /??
Accepted by.....
un
~uttntal Committee
on Graduate Students
oc9 5d
ELASTIC WAVE PROPAGATION
IN HOMOGENEOUS TRANSVERSELY ISOTROPIC MEDIUM
WITH SYMMETRY AXIS PARALLEL TO THE FREE SURFACE
by
Chi-fung Wang
Submitted to the Department of Earth and
Planetary Sciences on September
, 1973 in partial
fulfillment of the requirement for the degree of
Master of Science
ABSTRACT
Phase velocities of elastic wave propagation in a homogeneous transversely isotropic medium with symmetry axis
parallel to the free surface of a half space is investigated.
Approximate solutions of the problem of phase velocities of
Rayleigh, horizontally propagating P and SH waves is obtained
by means of perturbation method on the assumption that the
deviation of the elastic coefficients from isotropy is small.
In the case of horizontally propagating SV waves an exact
solution is obtained. The vertical lamination model approximating fracture zones and the Olivine model based on Francis'
hypothesis have been tested. The results derived from fracture
zone model showing small anisotropy fail to explain the
observed data. The Olivine model showing large azimuthal
variation of P and Rayleigh waves needed some modification in
such a way that the a axis of Olivine crystal will be distributed diffusely. Suitable choice of weighting functions
averaging the orientation of a a axis will give close agreement
between both observed and predicted P and Rayleigh waves.
Keiiti Aki
Thesis Supervisor:
Title: Professor of Geophysics
TABLE OF CONTENTS
Page
INTRODUCTION
4
STATEMENT OF THE PROBLEM AND EQUATION
OF MOTION
8
BOUNDARY CONDITION
17
THEORY OF THE SURFACE WAVES OF
RAYLEIGH TYPE
19
TRANSVERSE ISOTROPY AS THE LIMITING CASE
OF A LAMINATED MATERIAL
31
OLIVINE MODEL BASED ON THE FRANCIS'
SUGGESTION
39
PERTURBATION TECHNIQUE FOR THE P-WAVE
VELOCITY
52
ANISOTROPY OF S WAVES
59
TRANSVERSELY ISOTROPIC MEDIUM WITH THE
ORIENTATION OF SYMMETRY AXIS DIFFUSELY
DISTRIBUTED
62
CONCLUSION
84
REFERENCES
87
ACKNOWLEDGEMENT
-4Introduction
A great deal of attention has been given to the anisotropies of the propagation velocities of seismic waves in
connection with the investigation of the structure of the
Earth's crust and upper mantle.
An anisotropic medium is
characterized by the change of its elastic properties with
the direction.
In seismology, the transverse isotropy in
which the elastic properties remain invariant in the plane
perpendicular to the symmetry axis has been of the greatest
interest.
Stoneley (1949) is the first seismologist who discussed
surface and body wave propagation in a homogeneous transversely
isotropic half space with the symmetry axis normal to the free
surface.
Synge (1957) showed that the propagation of undamped
Rayleigh waves do not exist unless the symmetry axis is either
parallel or perpendicular to the free surface.
Bulchwald (1961)
discussed the waves radiated from a time-harmonic source and
used the method of Fourier double integrals to obtain the
equation for the velocity of Rayleigh waves which is the same
as that obtained by Stoneley.
A rapid variation of elastic constants with depth may
apparently have the same effect as an anisotropic medium on
the propagation of long waves.
Postma (1955) gave the explicit
formula for the five elastic constants of homogeneous transversely isotropic medium which is equivalent to a periodic,
-5isotropic two-layered medium in the long wave limit under the
restriction that the Lame's Constants are positive.
Backus
(1962) applied the averaging technique to the constitutive
relation and equation of motion to approximate an inhomogeneous
isotropic and transversely isotropic horizontally layered
medium by a long wave equivalent, but more slowly varying
transversely isotropic inhomogeneous medium in the direction
perpendicular to the layers.
Their theories will motivate us
to set up a model to approximate the crust in the Nazca Plate
discussed in this paper.
The anisotropy of the oceanic uppermantle beneath the
Mohorovicic discontinuity is first suggested by Hess (1964).
In the light of the observed results of the measurements of
P
velocities showing that low velocities perpendicular to
the fracture zones and high velocities parallel to them in the
region near the ridge axis in the East and North East Pacific,
he claimed that seismic anisotropy of the upper mantle beneath
the oceans results from a preferential alignment of Olivine
crystals and predicted that the b axis that tends to be
perpendicular to the fracture zones could explain the observed
anisotropy.
Francis (1969) suggested, on the other hand, that
the a axis of olivine crystals tend to point away from the
ridge axis and the b and c axes will be randomly oriented in
the vertical plane parallel to the ridge axis at the time that
-6the oceanic lithosphere was produced.
Backus
(1965) used perturbation technique to derive the
general form of the azimuthal dependence of the phase velocity
of the Pn waves as a function of the azimuth of the wave number
vector.
He showed that correct to the first order in pertur-
bation, the expression for the Pn wave phase velocity can be
written as
~9A 1
+ 36'P
+1 A%.Co2. -1
-1A+
4 Co.54 + A Sin4e
where the five coefficients A
are functions of the elastic
constants of the wave medium.
Later on, all the investigators
utilized the formula derived by Backus to interpret their
observed results in seismic refraction measurements (Raitt,
1969; Morris, 1969; Keen and Barret, 1971; Raitt, Shor, and
Morris, 1971).
Forsyth (1972) observed that there was a 2% azimuthal
variation of Rayleigh wave phase velocity in the region of
Nazca plate.
Smith and Dahlen (1972) combined Rayleigh's
principle and Backus' harmonic tensor decomposition to discuss
the effect of small anisotropy on the propagation of surface
waves of Rayleigh and Love type with the motivation of explaining Forsyth's observed value.
They also obtained the general
form of the dependence of surface wave dispersion on the azimuth
-76 of the horizontal wave number vector.
Much more detailed study of the propagation of surface
and body waves will be required in order to predict the
degree of anisotropy of the earth's crust and upper mantle.
However, it is worth adopting some simple models to compare
the theoretically calculated results for comparison with
observation.
In this paper, two such simple models will be
set up to explain Forsyth's observation and previously observed
Pn wave anisotropies.
For simplicity, we shall assume that
the media in both are vertically homogeneous.
The first model
will be a laminated mantle in which soft vertical layers
representing fracture zones are sandwiched alternately between
hard layers.
The second model will be a homogeneous half space
made of Olivine crystals in which the orientations of the b and
c axes are uniformly distributed over the vertical plane and the
a axis distributes with a certain distribution function with
respect to the normal to the ridge axis (Francis, 1969).
Both
models can be reduced to a homogeneous transversely isotropic
half space with symmetry axis parallel to the free surface.
In this case, the problem of the surface waves of Rayleigh
type can be solved by means of the separation of the equations
of motion and boundary conditions under the restriction of
neglecting the displacement of SH type which is of the second
order in perturbation.
The powerful procedure analogous to
-8that of Stoneley will be used in both the lamination model
and the olivine upper mantle model.
In order to relate the
Pn wave anisotropy and Rayleigh wave anisotropy, body waves
will also be considered in the equivalent homogeneous
transversely isotropic medium derived from the second model
and some of the necessary averaging technique will be taken
into account.
The purpose of this paper is to present the
theory based on these two models to explain the observed P n
wave anisotropy and Rayleigh wave anisotropy simultaneously.
2.
Statement of the problem and equation of motion
Our analysis is primarily concerned with the propagation
of surface waves of Rayleigh type in two simple earth models:
one is the model of 'fracture zone' in which soft vertical
layers forming the fracture zones are sandwiched in the normal
oceanic lithosphere, and the other half-space composed of
olivine crystal in which the a axis of the olivine crystal
tends to become perpendicular to the ridge axis and parallel
to the horizontal plane, and the b and c axes orient randomly
in the plane perpendicular to the a axis.
In either case,
the problem is reduced to finding the phase velocity of the
surface waves of Rayleigh type in a homogeneous transversely
isotropic medium, whose axis of symmetry is perpendicular to
the ridge axis and parallel to the free surface.
-9-
Let
(-
)
1|,
be an orthonormal coordinate frame
oriented in the right-hand sense at a point of the free surface
with T
pointing in the direction of the axis of symmetry,
directed vertically downward into the medium.
To obtain
expressions convenient for comparison with observations, we
have to choose another orthonormal coordinate frame
1,
q',3')
obtained by rotating
( i,,
such a way that
become parallel to a vertical plane
ej
about the vertical axis in
containing the wave number vector.
In the notation of Grant and West (page 27),
the stress
components and strain components in a transversely isotropic
body are related by the following constitutive relation.
?uL
0
0
0
0
Px.
()4+4)
0
0
0
o
0
0
0
0
e.
2.)
(2-1)
Because of the existence of strain-energy function, there
exists the symmetry of the elastic coefficients
- C=
and hence the elastic constant tensor of the medium in our
-10problem can be written as the linear combination of tetradics
(Morse and Fashbach, page 72) in the following form,
C
AV~e
+ A11e~;
A.AAL4j
24-
(
+
)(~
il
+
J~~~
viCiSZi+
+(~)
)
i
±
(2-2)
Suppose that the departure from isotropy is sufficiently
small, we may assume that the elastic constant tensor
the medium is perturbed from an isotropic tensor t
t
~
+
where S$
of
to
is small compared to
Hence
we take
C~(
+
+~)~
+iv~i
(2-3)
4q +
where
.-4 1
htAfi,=
15+ in fl)
and
Next, we shall introduce the infinitesimal displacement
E( t)=%(L
+
(t.
+
4
3
arising from the wave
propagation with the wave number vector
-11then the strain dyadic tensor is
~(zi-i.w~z
where
a
T
denotes the transpose, and the stress dyadic tensor
is given by
or
where
(2-5)
+~r
~~
,~
z.~)(2-6)'
is the operator
which transforms the displacement fields
~(()
=
LL+(4..3
3
into the stress fields.
7 7 C3
In the discussion of the propagation of surface waves
of Rayleigh type, we are mostly concerned with the regions
devoid of sources.
The initial approach is to obtain the source
free equation of motion appropriate to the medium of our problem
in terms of the spatial derivatives of the displacement components with respect to the orthonormal coordinate frame
and then use the method of displacement potentials
for plane waves and the prescribed boundary condition at free
surface to obtain the modified Rayleigh's equation and
determine whether surface waves of Rayleigh type.can or cannot
-12exist in such a medium.
A simple means to reach our goal is to
express every vector and tensor in terms of their components
with respect to the frame
w%',
' 73O.
In homogeneous source-free regions, for time-harmonic
displacement fields with exp (-Cot ) dependence,
and assume
that plane wave solutions of the type exp
are admissible, then in the equation of motion,
be replaced by
=
replaced by
is
=
and
-w
+
3'
where
=
the position vector and
9
can
can be
X A=
=
'
(A", 2.,3)
The equation of motion in a medium of density
f
on the assumption of infinitesimal deformation, and in
the absence of body force is
(2-7)
-/"
1)3
or
Aj.)~
1z
-
'4
3~(-)
Our final step in obtaining the equation of motion
is
to express CA?>AAg
AD ), /1
)
A
.
9
in
terms of five elastic constants
through the coordinate transformation
-13-
C0SO 41
S ineSr,'
+ cose5'
(2-8)
and
Co5e
+ Sene
-t Cose
(2-9)
3
s-i
Note that the tensor C0 in
44
0
is
S$
an
isotropic tensor, which can be written as
(2-10)
+
In case of
the equation of motion is reduced
of course to the equation for an isotropic body
[(I,]
(A~~44~ 1±A ~3
0
At W- L)
%21
(2-11)
0
LL
O
lt
2
-14In a homogeneous transversely isotropic medium, the
departure from isotropy is due to the existence of three
parameters
of
6>, 6f and
.
The absence of symmetry
in the horizontal plane requires some complicated
SC
algebra to obtain the right hand side of (2-7) in terms of
the five elastic constants
Al
.
the coordinate frame
I
-(COS
f
,
li.
,
At
Using (2-8),
and
9
in
(2-9) and
4- (Exa
(A
+ (L3e 3,
-Sk($noU3.) (5)
Qw j)
+
('3
(2-12)
then after some lengthy calculation, (2-7) becomes
Ile
0
)VI (71) At','(*)
jG4V
Ir, Fa
A04
& 0-) A Q70
41 (Q,9.447i) A;(70.11
AIz £A12&',
(viI
'13J
.3
~
$$ph)
hwar)
{(t3
(2-13)
A33)
where
(AOA-)
IJ3
Al (h) A'O (Q)
h (Vr)
A51 (7T) A*aE()
Ai3 (V)
A44? +
(2-14)
-15and
is a 3x3 symmetric matrix with elements
. A,1(v)=Au4
r)
(-4x5irne (as e-25A.S
S
E(=
~3MG2
.)
COce+ 65SC52ce)d
+$)c.
(2-15)
Equation (2-13) constitutes a set of three coupled
simultaneous linear partial differential equations in three
unknowns
(
(tI
represent
d=
t
becomes
i2. and
C a.,
C3 .
To solve this set, we first
n(
&d-.3
as consisting of part due to
The displacement in (2-j3)
plus a perturbed part.
M
=
+
TCm
+
+
IM
-
(2-16)
The zero subscript denotes the value of ' in isotropic medium
with ,Al and Aq as Lame's constants, and the i(i>Q)
subscript
denotes the perturbed portion of - which is of i-th order in
and SA .
term in
(-
than zero in
-=
If
2
SA
S)
we assume that
=0
, then every
is of order higher
V.--14,
and
'
*
We note also that the
-16-
elements
Ami (W)+$Am(w)=A4)+6Aw(v2); A357()
Am (+)
and
+
(7)
terms involving
6,k 3x&
$9.
=
A"(9)+6A3X(W)
3))±SAg
Neglecting all
(L which are of order higher than one in
we get
(All +&1 (Li + (Ao(7)+SAt~ 3
Ait~ + (A2L(wr)+ SA2w(r) (L-+ 6~v
Z)+ 6A,(V))
333
(2-17)
The second equation in (2-17) offers explanation how horizontal
transverse displacement (
can be generated by (LI and
U3
through the perturbation of the elastic properties of the
medium.
On the other hand, the motion governing
U.
k3
and
can be discussed in two dimensions
(X',
containing the wave number vector.
Therefore, only the first
x3')- the vertical plane
and third equations of (2-17) are needed to discuss the
propagation of P-waves, SV-waves and surface waves of Rayleigh
type; whereas, in considering the propagation of SH-waves,
(
and (L3 are regarded at least of the first order in
perturbation and the second equation of (2-17) becomes
t
fAA20 (W) +
2
.427
(2-18)
-173.
Boundary condition
The boundary condition of our problem is the vanishing
of all the components of stress working at free surface X3 =0
The combination of this condition and the condition at
infinity:
Z=O at X3'= Oo will determine the property of Free
Rayleigh waves.
The stress working on the horizontal plane
is
3'{J
+
3)
J(3-1)
For the isotropic part, we have the well known result:
Next, we shall consider the perturbation term of (3-1).
(3-2)
From
(2-4) and (2-6), we have
{f3
3
(3-3)
-18Applying (2-9) and (2-12) to (3-3), we finally find
Coe5(c
5'(('g)$}5=
33
- 5,9 So(se
+ $1?A(Coo
Q--S
0
(4. c) 5
Sn9
) '
e+oso,)
- Sheresend
1
S
(3-4)
When we are interested in the motion described by Cian
only,
(
and
6A(.
l C3
may be neglected for the reason
discussed in relation to (2-17) and hence (3-4) can be
written approximately as
(3-5)
Then the boundary condition at free surface
=o
becomes
(3-6)
This relation plus the condition at infinity that the energy
will decay to zero is sufficient to determine the motion of
free surface waves.
When
C
(3-6)
reduces to the
-19boundary condition for isotropic body.
A10141t
+
(,4
3 +-
4.
0
(~+A,~L3
(3-7)
(k= 0
Theory of the surface waves of Rayleigh type
In order to obtain the phase velocity of waves analogous
to Rayleigh waves, we shall apply the method of plane waves
due to Stoneley (1949) to our modified equation of motion
and boundary condition at free surface.
Recall that in Section 2 the first and the third
equations in (2-17) are sufficient to discuss P-waves,
SV-waves and surface waves of Rayleigh type approximately
and the second equation will be ignored for the present so
that the equations of motion expressed in terms of the components
in the coordinate frame
IFd?(L
3where
(F iL),
(tt
{
'
(
( U)
are
(4-1)
-20-
A~
Gwiiki+
~
L =
+59e
C =
A
2
GjY~~5e +F~5~)i~
(4-2)
Now we set
(4-3)
or
Assume that waves are plane of sinusoidal disturbance,
the displacement must have exp(--.2iid) dependence, where
is the circular frequency.
Wo
Then we can write (4-1) as
(4-4)
Note that we no longer have in general purely
compressional waves or purely rotational waves as in the
isotropic medium.
7 and 8.
The details will be described in Sections
In order to investigate the surface waves of
Rayleigh type, it is convenient to derive the modified
Rayleigh's equation by means of displacement potentials.
-21Considering homogeneous plane waves with the phase velocity
c, we assume that
(4-5)
oo
and insert them into (4-1) and (4-3) to get the following
equations.
i (,rc4)-0A4
(f~2 2L
+ (F+2±Ly+]
+
[L 3X-+
(rc-A+FiL)
+ (L--Cc )L- =
cF)
-
o
(4-6)
or
9 = o,5 T
A
4h=\
= =o,
the equations of motion
are reduced to those on unperturbed isotropic medium with
Lame's constants AIj and ,e
,
and (4-6) becomes
+
+7
)
(4-7)
which correspond to equations
and
)(."i+
-
waves in isotropic medium, where
+i
+
0
--
-f
for inhomogeneous plane
C
~
and
-
-22We assume further that
+())=
-
_
0.4p(
3 1)
From (4-7) and (4-8)
are constants.
where
yl='
and
we have
CX-A -
Ae
f
f' (Ft 2-L
c--2L
c)
Q
-
=
.'(c-iiF+L-
L
-
L-CtF =o
(4-9)
In order that inhomogeneous plane waves may exist,
there must be non-trivial solutions
0 and
)' of
(4-9).
This
condition is
(1-g{L
-
C
A
+ ( (fcr-A) C Or + L(fes -t) + (FPL)
(fc_--4)(c1-L
0
In equation (4-10), the solutions
(4-10)
(=:J
have nothing to do
with the motion of surface waves of Rayleigh type.
solve for
in (4-10) to give the values
LC
We can
-23-
(4-11)
Lw
<=
where
For
6
=O,5-r
~
C1 -A) + L.(fe -L
(F
or isotropic medium with
Ali)
.l
as
Lame's constants, they reduce to
g;~=~
~-
7fc7
.
There are two roots 9
of equation (4-10) , and
and [
hence the solutions of (4-7) must be of the form,
E
~=
4f
According to (4-9),
*&)=9,og(14x
(-it
l3') + .Y2X (-&l
ir2X5 )
(4-12)
(4-12) can be written as
-- $.o
L
ex0
(4-13)
-24in which both g, and fsmust be negative in order to satisfy
the condition at infinity, where
-
f.c -A + 3&(F-L)
fc7- - A t F t L + L,,:
( ,
Now that we have (4-13) which expresses the characteristic
of surface waves, we are going to put them into the boundary
conditions at free surface given by (3-6) to obtain the modified
Rayleigh's equation.
Since we are not interested in
, for
the time being, we only write the condition in the following way.
+
(~±±~)
.i6.~3
+
~~
~-i-~
(~0
~3 ~
(4-14)
In terms of compressional and rotational potentials, (4-14)
becomes
Az bi J + 32. ' 11)( = 0
On substituting (4-13) into (4-15), we obtain
(4-15)
-25~A x ,
tQP(fg.i) (N~24g
/Nil
Ae.
(I
~
(4-16)
For non-vanishing
4,
and
4,
to exist, the following
condition must be satisfied.
1k
N
(4-17)
It can be shown
which is the modified Rayleigh's equation.
that in an isotropic medium, this equation is reduced to the
ordinary Rayleigh's equation
-
for arbitrary Poisson's ratio where C
:
-
and CS are phase
velocities of P wave and S wave respectively.
Let us look at the elements of (4-17) more explicitly
and find that
-26-
(frc'-A)
"no
(~f
c-A
Co
+ if"CAo +24 +
+)N4+,q +
<&A+b)CosLe+
<-h(AII+Al
+GA+ 9)Gs5'e )
C +Al
4
+ 0 A+)Cae
+
a
(A, H6 Cose)
+-t
?
+A1 Cos'0 )
(t 2 -- A +.u+ All + (b +S9) Cos's ± (a +±COS+LO)v)L
(Cfe4)
(fi-()
N
{(ki
_
(t+~;)
A
+
( (peC -A)
k fc '&-A
- .
+,Al
+.A
(,
+
+ ;A (0s5e6)
(6A+ 6S)Co5o
+ (A, +P5&oAe)V
)
(4-18)
where
A=
h
+2.fA + S A
+
is
indicated in
Coe
(4-2)
C(ots ) +2.e5A Coslo +45p S1 M]26
-27From the above results, we find a common factor
(fC'-A) + Al tAt + ( A+'9)
in the column of (4-17).
Cos+ ± (Ali +9 Cose) &
From the expression of
we see that the matrix in (4-17) has rank 1 if we set
and hence there must be a common factor
the left hand side of (4-17).
After removing the factor
whereAF-and L are expressed in
-
[N5) (Alt2t)
+
f c-A)
(A+k) O+2
+(A )
+9(A+4-)(r
+4
(A+4(a
t
on
(4-2),
we get
(A
u)&I±A
F.
) (Y 0 -A)
-A)(c-) 5'(f'A
=o
(4)(f-A)
(4-19)
-28where
From (4-11), we recall that
(4-20)
+
A
+
+
Substituting (4-20) and (4-21) into
C
AL)
(4-19) and after some
arrangement, the modified Rayleigh's equation is finally
obtained.
(4-21)
-29-
A+,
~
0
(,(+2L)
X%~ -2"A) +
(AIj+6F)2-
(4-22)
where
X=fC
For small anisotropy, Al%4L.0O
in the medium under consideration.
and hence there must be a root of
interval
(
>0
X
,4f .f%)
and ( i4 by)(A
We have
F 00=O
and P Auf f)>o
in the open
For real roots, we must have
which is equivalent to
<Al,+IAR+SA
RO)<0
(X-,-k))(4
X>kAt+L , X >A+2/44fA
X<fAtjk
F(X) only takes real roots for the second case.
but
Rationalizing equation (4-22), we obtain a polynomial
equation in
4L
--
eie
(4iL
7j'%
of degree 3,
SL i
{All:& Y.(4~
+~~~~
-30-
+~~
(-t-L,(I+
~2
(4-23)
2
where
for the poisson solid satisfying
' 2 -.--- (+
Aw) -)
where
0
2J
-2+
7=
.
,
we have
2
(4-24)
+
(j+ ).3
Both of (4-23) and (4-24) are the modified Rayleigh's
equation from which we can determine the phase velocity as a
function of the azimuth of wave number vector.
In the
following sections, we shall calculate the ratio of the phase
velocity to
corresponding to
from (4-23)
0* I*,'
=
=C
and (4-24)
in ten directions
and 90* for the two
models; the fracture zone model and the olivine model
discussed in the introduction.
-31-
5.
Transverse isotropy as the limiting case of a laminated material
The motivation of our choice of approximating the oceanic
lithosphere by a laminated model comes from the conjecture that
the fracture zones may be regarded as the soft thin layers
sandwiched in the Normal Oceanic lithosphere and in welded
contact with it.
Postma (1955) has shown that in the long
wave limit, a medium consisting of alternating plane parallel
isotropic layers of two different elastic properties can be
approximated by a homogeneous transversely isotropic medium.
Since Forsyth found the anisotropy for Rayleigh waves in the
Nazca plate up to wave lengths of 400 km, the long wave
assumption may be justified.
Consider a laminated half space, in which vertical thin
layers with thickness h, and Lame's constants
/s,141
sandwiched in a half space with Lame's constants
and thickness
ho
as shown in Figure
orthonormal coordinate frame
f(i,
1.
%A3
are
a, /d
Let us choose an
at some point on
the free surface such that f is the horizontal unit vector
normal to the plane of lamination, and i
is the unit vector
directed vertically downward into the medium.
of the frame is also indicated in Figure 1.
The geometry
Following Postma
(1955), we choose a fraction of their thickness as the weighting
function to average all the stress and strain components in
these two different isotropic layers, finally the averaged
stress and strain relation is obtained.
Comparing this
-32-
X074 L
X
T
7/'
V
A
e2
-
+-
Figure 1.
h0
--
A
Coordinate frame fI, Fzf3
laminated half-space.
attached to a
@i is the horizontal
unit vector normal to the interfaces.
the vertical unit vector and
2
A
is
-33result with the constitutive relation in homogeneous transversely isotropic medium with symmetry axis parallel to E
we find
(A o +
2/J
A44)t+
'+
it4-
(Ao+2Ao) t
where,
(
0)
(5-1)
A-
There are five elastic constants of the transversely
isotropic body equivalent to the laminated body at long wave
-34For simplicity, we shall assume Ie =Ao and A=/&t
C(e)
Then the ratio
in equation (4-24) was computed
limit.
y(9)
'
for
50
at 100 interval for several cases of J\=
and a fixed rigidity ratio.
2
for
The result is given in Table 1
For 9300,
which is the value of
()
=
134.o2.
in the isotropic Poisson's
solid.
We see from Table 1 that the variation of
__
with the azimuth of the propagation is small compared with
the 2% anisotropy observed by Forsyth for Rayleigh waves in
the Nazca plate.
The set of parameters
and
which give the required 2% anisotropy were obtained by the
following procedure.
For Poisson's solid, we assume that
Ao=Ao and
=
Then equation (5-1) becomes
S+
(5-2)
/A..
+(5-2)
-35Table 1. Y(e)
calculated for the laminated model in
= .21.16/16
which
A)." +0a
5
5
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
10
10
15
15
15
15
15
15
15
15
15
15
(Degree)
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
y(O)
0.9131
0.9132
0.9136
0.9143
0.9153
0.9164
0.9175
0.9185
0.9191
0.9194
0.9169
0.9170
0.9171
0.9174
0.9178
0.9182
0.9187
0.9190
0.9193
0.9194
0.9176
0.9178
0.9178
0.9180
0.9182
0.9185
0.9189
0.9191
0.9193
0.9194
-36from which we have
3&tAoA
_L
1
V-All
/I& +c~
/4_
.- iA1A q AI
1
4.v
'cA+
/'.AA
-A
A- (O-A0A,+(A-A)2
(I -
*
-
(5-3)
and hence
(4 t a,4,)d~ + a,4.)
54+A (A
(5-4)
)..
For given values of
(5-4) is reduced to the homogeneous quadratic equation
(-
)
+ a(
i+ --
( 4*
+ (+A)
) =o
-
(2
(5-5)
-37Table 2.
calculated for laminated model compared
(W=(Y
with observed by Forsyth (1972).
@ (Degree)
10
20
70
Calculated
Observed
0.899
0.900
0.901
0.903
0.906
0.910
0.913
0.917
0.919
0.919
0.901
0.902
0.903
0.905
0.909
0.912
0.915
0.917
0.919
0.919
-38Table 3.
);
and
which give the
$
Model parameters
observed anisotropy for Rayleigh wave phase velocity.
1
2
3
4
5
6
7
8
9
10
12
15
20
30
40
50
60
70
80
90
100
1.599
1.634
1.704
1.780
1.854
1.930
2.000
2.070
2.140
2.208
2.342
2.538
2.854
3.460
4.048
4.625
5.195
5.760
6.321
6.800
7.440
-39in which the root Q
)
must be greater than 1.
On the
other hand, from the requirement of 2% anisotropy, in
Rayleigh velocity the values of
.- oAgg 84.3
-O=-283
are required from the first
order perturbation of Rayleigh equation (4.24) in 4.)S, JA
and
.
--
By successive approximation, we finally obtain
=
034-5ET
which gives nearly 2%
anisotropy in Rayleigh wave phase velocity.
of
The variation
with 0 are given for the
final solution together with the observed data as shown in
Table 2.
Thus, the observed 2% anisotropy in Rayleigh waves
require the anisotropy parameter
From equation
as much as 0,o5I.T30
(5-4), one can find the combination of
parameters for the fracture zone model which give this
value for
.
The result is given in Table 3.
For
example, if the thickness ratio is 10:1, the rigidity
ratio more than 2:1 is required.
We consider the result
rather unreasonable and therefore conclude that the fracture
zone model is unacceptable.
6.
Olivine model based on the Francis' suggestion
Observed anisotropy of the upper mantle in the Pacific
Ocean appear to agree with the hypothesis that maximum velocity
-40is in the direction of sea-floor spreading.
Since Fracture zone
model is inadequate, an alternative Olivine model will now
be tested.
In this model, we can include the measurements for
short period P wave anisotropy simultaneously.
In this section,
we shall approximate the upper mantle by a half space made of
an aggregate of olivine grains in which the crystallographic a
axis lies horizontally in the direction perpendicular to the
ridge axis, and the orientation of the b and c axes are randomly
and uniformly distributed over a vertical plane parallel to the
ridge axis.
Let
,
E3
be an orthonormal frame field which is
parallel everywhere with i'
and
F,"
in the direction of a axis,
F"
in the directions of the b and c axes respectively of
a single olivine crystal, then the constitutive equation for
this grain is described by nine elastic constants as
(
c3
C.CX%
Cis
(C
00
_61010 0
o
Cci
,
C1 3
0
0
0
0
240
0
0
0
0
0
0
0
0
o
0
q
0
C
O2.(e6
6
(6-1)
-41from which the elastic constant tensor of this single crystal
can be written as
+
where G
/±.
C,%('
and e"
1
~'+
-W)+
are the components of the stress dyadic
'-=
AAs
and strain dyadic
Let
?
respectively.
be another orthonormal coordinate frame at a
point of the free surface of the half space such that
is the horizontal vector in the direction perpendicular to the
ridge axis and
medium.
is directed vertically downward into the
According to Francis' hypothesis, we may assume
first and hence these two frames are related by
a
and
rtS.
+ cs
(6-3)
-42-
ee=2
2
e3
e3" e3
Figure 2.
Coordinate frames
(ti,
e!,1
j
and
{
' E:;
~ is parallel to the free surface and normal
to the ridge axis.
E is downward unit vector
a
into the half-space.
3 and
~".
is the angle between
The plane spanned by Ez and 9
is the same as the plane spanned by
8
'and
'
is the azimuthal angle measuring the rotation
around the vertical
*
.
-43-
Coa
(6-4)
The geometry of this transformation is shown in Figure 2.
A great deal of recent work indicates that preferred
grain oreintation is probably the most important factor in
producing anisotropy in a compact aggregate (e.g., Babuska,
1968; Kumazawa, 1964; Klima and Babuska, 1968; Crosson and
Lin, 1971), and hence we shall neglect the interface structure
between the grains in our model.
In order to predict the
elastic properties of the model in which the b and c axes
orient randomly in the plane perpendicular to the a axis
(or g), we shall go through the procedure according to Voigt,
Reuss and Voigt-Reuss-Hill approximation with the spatial
average replaced by averaging the single crystal property
with respect to rotation around the a axis.
If the components C
in (6-2) which form a tensor of
fourth order are transformed into the components of C
the coordinate frame
C A3i
7"
R
in
and we write
.
Then the components
(4) are functions in the angular variable
e)
-44according to the transformation represented by the rotation
(6-3).
In the Voigt's scheme it is assumed that the strain
field induced is uniform throughout the aggregate, then we
consider an equivalent uniform elastic body which would produce
the averaged stress for the given uniform strain.
The stress-
strain relation in this set of grains can be written as
C(
all-(a)
'(6-5)
.3
Z-A
;
'A
where
*
in (6-2).
The averaged stress field in the whole aggregate is
is the elastic constant tensor
then given by
K
9-
The Voigt's averaged elastic constant tensor is
(6-6)
,
After
the evaluation of each integrals with components of c(a)
as
integrands
<C>v
c '
+ CX'I'z x y&; + C3 3A"3r
+F
LT) 4- C O (VS3+'f~'f)T
T
+ S" tr L§++
(6-7)
-45where
CY= C+
CIII= C113= 1:CI-+C3
(ct +C33) -(-44c4-4c
cyt
C3 3)
%~~~C
#G+
C
3
L= +(c
3 +c+)
C
+
(6-8)
According to equation (6-7),
becomes the elastic constant
tensor of a homogeneous transversely isotropic medium with
%
as the symmetry axis in which the matrix representation of this
tensor is
CY
CY
CVa
c,
cit
0
c"
3
cV
3
0
0
0
0
C4
0
0
0
0
0
0
0
0
o
e
o
0
aCSY
0
For our convenience, we shall adopt the notation in (2-1) and
write
-46-
el I
Cy
0
0
0
q2.
C3
o
0
0
Cra C0
C0a
0
0
0
2.C4
0
0
0
0
o24o6
0
o
0
0
o
AL"
W+
I~aA)
"'Iv
V (AI
+2/ kVi) O0
0
0
0
YR)I
0
0
0
0
0
o
0
2c4
o
0
o
0
1 V0
o69
(6-9)
or
V
94
=
.C
Ogv
S
6
(6-9)'
In the Reuss scheme, it is assumed that stress is uniform
throughout the whole aggregate, then we consider an equivalent
uniform elastic body which would produce the average strain
for the given uniform stress.
Hence
(6-10)
where the compliance tetradic tG) is the inverse of
tensor
cQ)
when we regard them as the symmetric operators on
the 6-dimensional tensor space.
-47In the notation of matrix in the frame
(
can
be represented by the matrix
s,S13 0 0 0
Ss
Sla. S
(s]=
ftW
513 -%3533 *
0 0
4
0
0 0 0 0
0
0 0
0 o
lS3
C"l
C13
where (C
St3 0 0 0
2&2
eta 4O0
0 0
Cual 023 0
0 0
0 0
0
C2.3 C33 0
0 0
0
0
0
0
0
0
0
:L:
3
0C
O
Pc
C55 0
(6-11)
is the inverse of the matrix (C) represented in (6-1)
and hence in tensor notation,&A) can be written as
+3
±)+~
'+
(;e '3+ Z3';r)) OP41
+~
~~
53
C4
#3#
cf- 't1'±
e) (Xte~)
(; jjez' t
(6-12)
-48Following the same procedure as that in Voigt's scheme, we
find that the Reuss average compliance tensor becomes
+ SQ(3 )
+
+
+e)
sr(3+ )((+)
A(6-1'
t
3)(L2)
(6-13)
in which
Si,
4.= 3.j= o,37.
it=
SA=
3(
ss(S
=
( Sa + Sa3 )
+
0%25,5x3
+
o,554
(512.+313)
-33)=- 0,%12.-(S
o,s (Ss--
+ St)
+ S5)-4,5
ss
4+
(6-14)
-49The matrix representations of the elastic compliance tensor
and stiffness tensor are related by (6-11) and hence the
matrix representation of the Reuss average elastic stiffness
tensor
+tT1l('l+
4-CIL (+
is obtained.
±
ei)
Va
)
For any values of C
-S(
in (6-2), we find that
still represents the transverse isotropy with
as the symmetry axis.
In the notation of the preceding
sections also, we set
KA---=C K
E
Wt(6-16)
(6-15)
Ft
-50-
The Voigt-Reuss-Hill (VRH) approximation, the, arithmetic
mean of the elastic constants calculated according to the
Voigt and Reuss schemes, is given by
C;!= Os(C4
+Cj)
Now we shall go back to the discussion of the phase velocities
of the propagation of surface waves of Rayleigh type on the
model with elastic constants calculated from these three schemes.
Let us choose Olivine (93% Fosterite) as an example, the
density and the elastic constants are as follows:
dcei.s;4/ f = 3.3110 3%KXA
C,= 3.23O M6.
Cal= 1,376 oN b
Qis=-.f 2,3 5.1 M b.
O0n3= e-T5
0e M b.
C4 =0,64G2.Mb.
Cm= -. 805 rib.
Ca= o.T
Mb.
According to the foregoing procedures, the five elastic
constants of these three schemes are
Voigt:
A= 7647T5 PMb
=
0,6+3TS Kb
4.:: o, 63o oooM
AV= A.2T300 ML
PV= oq,8545o NMb.
(6-17)
-51Table 4..Y(e)=
"Olivin
6
0
10
20
30
40
50
60
70
80
90
<
computed for three schemes of average for the
Model" by Francis.
(Degree) Voigt value
1.0184
1.0153
1.0063
0.9924
0.9755
0.9580
0.9527
0.9313
0.9246
0.9224
Reuss value
VRH value
1.0254
1.0220
1.0124
0.9975
0,9793
0.9604
0.9435
0.9308
0.9231
0.9205
1.0220
1.0187
1.0093
0.9950
0.9774
0.9592
0.9431
0.9311
0.9239
0.9215
-52Reuss:
0633500 M6
O
So,661440 Mb
6
M6.
,o63638
At
L.0
1.265040 Mb
0,7854 t3 Mb.
(6-18)
VRH:
=
0,'4
'2\ 37 Mb.
-o=
-6 682. o m b
o,663133 Mb.
=48543+
Mb
(6-19)
Substituting all the values of these three schemes into
equation (4-23), we obtain the calculated values of the ratio
=
200,
c(e)
...
in ten directions corresponding to e = 0*,
, 9 0 * as given in Table 4.
10,
The results show that the
predicted anisotropy is much stronger than the observed 2%.
In order to obtain agreement with the observation, we shall
diffusely distribute the a axis around the normal to the vertical
plane along the ridge.
In the following sections, we shall find
such a distribution of a axis that will explain both P and
Rayleigh wave anisotropy simultaneously.
7.
Perturbation technique for the P-wave velocity
Accumulating evidences from refraction measurements
-53support strong anisotropy in the velocity of horizontally
propagating P waves in the top of the Oceanic Upper Mantle.
The propagation of body waves through an anisotropic media
may be described by three phase velocities obtained from
the dispersion relation.
As already mentioned, purely
longitudinal waves and purely transverse waves can no longer
exist in a generally anisotropic medium.
We shall, at this
point, treat the case of small anisotropy analogous to the
work of Backus (1965) by means of the perturbation technique
in dealing with the coupled body waves.
We shall assume that the wave number vector is horizontal,
and the model will be the half space composed of aggregate
of Olivine grains discussed in Section 6 which can be specified
as a homogeneous transversely isotropic medium having the
symmetry axis in the horizontal direction
,
In considering
.
the phase velocities of the propagating waves, plane wave
approach could be an appropriate one.
If
C-(r)
is
a plane wave of dependence exp {(
then it will satisfy equation (2-7).
let
c
be the phase velocity and
Define
3 (T)
7
f -
fX
=
(7)
be the positive definite symmetric tensor of rank 2,
where
(
3(7))
is the contraction of
elastic constant tensor defined by
-t))
())
-54-
then equation (2-7) becomes
B()
M
()=Ca7E7
(7_1)
The three eigenvalues of
67) which are squared phase
velocities correspond to the polarizations of three body
waves.
It is clear that the tensor S(7) can be separated
into a part 4(7) which describes the propagation in isotropic
medium with Lame's constants
A u,
*
and another part
b
()
which describes the effect
due to the anisotropy, namely
(7-2)
where
=
the is
v,3
pc))
C31'
(7 2)1
From the isotropic tensor Cc given by (2-3), we can write
-55-
Lr(7-3)
5o
Hence, we have the degenerate case for an isotropic medium in
which the polarizations of the longitudinal waves and transverse
waves are uncoupled.
In this section, we consider the propagation of P waves,
SV, and SH waves will be discussed in the next section.
that the unit vector
-P1
section 2 is precisely 7
waves.
4
Recall
in the orthonormal frame defined in
for horizontally propagating body
As we have already seen that from Se given in (2-4),
is not eigenvector of
and hence the polarization is
no longer longitudinal for P wave.
However, the perturbation
theory (Backus, 1965) which is nondegenerate for P waves
provides us with the information on small anisotropy.
write the eigenvalue of
()
as
the first order in perturbation
from
+ S;
then,
If we
correct to
-56-
Z(75)
)
and (2-15)
A&(V)
where
replaced by
gCZt=os
=
are given by (2-15)
with
Vr
We finally have
(S^(t+Sie)+±.54.(cs + 4
1936
(7-6)
which is the deviation of the squared phase velocity as a
function of the azimuth of P wave propagation, which is reducible
to the general form provided by Backus.
In general, when we want to evaluate the phase velocity of
the propagation of P waves in any weakly anisotropic medium, we
may impose the condition that wave number vector is in the
direction of the eigenvector (to be more precise, the polarization
of the particle motion is along the direction of the wave number
vector) regardless of its coupling with other modes.
The Pn velocities calculated by the formula
C
)+(T)Cos'e
H(-3)
+ 2ACos
e
(7-7)
.+4S)
-57Table 5A.
6
0
10
20
30
40
50
60
70
80
90
P, SV and SH wave phase velocities computed for
Voigt scheme for the "Olivine Model" by Francis.
(Degree)
P
Ij"/g
Ic)a)(
9.8876
9.7984
9.5509
9.1994
8.8195
8.4856
8.2452
8.1065
8.0453
8.0295
4.8705
4.8602
4.8303
4.7784
4.7269
4.6652
4.6065
4.5581
4.5262
4.5150
4.8705
4.8933
4.9542
5.0146
5.0560
5.0560
5.0146
4.9542
4.8933
4.8705
-58Table 5B.
P, SV, and SH wave phase velocities computed for
Reuss Scheme for the "Olivine Model" by Francis.
G(Degree)
Table 5C.
0
10
20
30
40
50
60
70
80
90
)
L5~L
9.6814
9.5918
9.3419
8.9836
8.5895
8.2330
7.9641
7.7975
7.7157
7.6925
0
10
20
30
40
50
60
70
80
90
6
~
(xyAe()
4.8705
4.8589
4.8253
4.7734
4.7090
4.6400
4.5731
4.5183
4.4822
4.4696
4.8705
4.8859
4.9249
4.9688
4.9972
4.9972
4.9688
4.9249
4.8859
4.8705
P, SV, and SH wave phase velocities computed for
VRH schemes for the "Olivine Model" by Francis.
(Degree)
CA #kg'C)
9.7850
9.6957
9.4470
9.0921
8.7052
8.3453
8.1059
7.9535
7.8822
7.8628
4.8752
4.8595
4.8280
4.7760
4.7180
4.6524
4.5898
4.5382
4.5042
4.4924
4.8705
4.8896
4.9377
4.9917
5.0267
5.0267
4.9917
4.9377
4.8896
4.8705
-59for 10 directions 0
0*, 100, ...
, 9 0 * for the 'Olivine model
with averaged elastic constants of three schemes mentioned in
the last section are tabulated in Table 5.
all too large along the direction of
7 .
These values are
We must conclude
that the Olivine Upper Mantle model in which the orientation
of the a axis of all the Olivine grains lies in the horizontal
direction perpendicular to the ridge axis cannot explain the
observed anisotropy.
It is again necessary to diffusely
distribute the orientation of the a axis.
8.
Anisotropy of s waves
From (7-3) we see that the eigenvalue
)
of
,(i)
is degenerate with a two dimensional eignespace normal to k
generated by (
and
'.
0(11) .. .EX2± st(4).
If we write the eignevalues of
in terms of (~)~~(~.
for waves of SV and SH type respectively, then again to
the first order in perturbation by imposing the condition
that
3 and l are the eigenvectors corresponding to the
waves of SV and SP type respectively, we find, following
Backus, that
69
following matrix
and $4) are the eigenvalues of the
-60These two eigenvalues can be obtained by choosing two
orthonormal vectors which generate the same eigenspace
normal to f/ and diagonalize the matrix in (8-1).
In the present case of transverse isotropy, the problem
is much simplified.
Synge (1957) has shown that in a
homogeneous transversely isotropic medium, there is an
eigenvector of 8(ft) normal to the symmetry axis and the wave
number vector.
Hence in the medium characterized by the
=
elastic tensor
+' r
with
'
and
$'
the form of (2-11) and (2-4) respectively,
eigenvector of
7(')
(')49'
expressed in
' is the
and the matrix in (8-1)
is diagonal, namely, the polarization of SV waves is purely
transverse.
For the waves of the SH type, the polarization
is no longer purely transverse.
of obtaining (8-1)
gives
C
:
However, the approximation
-
(;
'-
correct to the first order in perturbation.
As long as it
is permissible to impose the condition that the polarization
of SH waves is transverse, the azimuthal dependence of the
phase velocity C5N will be obtained.
Summarizing the foregoing remarks,
exactly for SV,
and
JC=K(C2?
(6
we have
Se--
('A
SC='
2.$%)3Z6(oS
correct to the first order in perturbation for SH waves.
In other words, the phase velocities C
written as
and CSH can be
-61-
.m.~/%I(C
~ ~
+a+All's4ah
~cs
(8-2)
+ A 5 'e
~r'6
(8-3)
The phase velocities CSH and Csv given by (8-2) and (8-3)
are also tabulated in Table 5 for the three averaging schemes
in (6-17),
(6-18) and (6-19).
They show that the azimuthal
dependence of CSH is weaker than C v!.
P waves exhibit much
greater azimuthal dependence than S waves.
The phase velocity
of SV is monotonically decreasing from e = 0*
to e = 900,
while that of SH waves is the smallest at 6 = 0* and 90* and
the largest at e = 450*.
These features should be helpful for
diagnosing the anisotropy of a transversely isotropic body.
-629.
Transversely isotropic medium with the orientation of
symmetry axis diffusely distributed.
So far, the phase velocities of Rayleigh waves in 'fracture
zone' model and 'Olivine' model computed in Section 5 and
Section 6 were unable to explain the observed data; the latter
gave too large azimuthal variation and the former too little.
Also, the calculated values of the phase velocities of P waves
did not fit the observation.
In order to explain observed
results, we shall modify the 'Olivine' model in such a way
that the a-axis is no longer strictly perpendicular to the
vertical plane along the ridge axis, but diffusely distributed
around the direction.
Since we found in Section 6 that the
bounds given by Voigt and Reuss averages are narrow enough,
we shall calculate only the Voigt average here assuming that
the strain is uniform.
On the assumption of uniform strain throughout the whole
medium, the averaging of the elastic constants is reduced to
the averaging of the equation of motion.
Before this average,
we have to perform the transformation of the coordinate frames.
EL4'1
%',' #,,,
Let
and
{fi,
'
be three
orthonormal frames having the same origin at a point of
the free surface of half space with the geometry described in
the following way:
is the horizontal unit vector normal to the ridge
-63axis, (
and
is the unit vector directed vertically into the medium
V is
obtained by setting
is obtained by rotating frame
an angle
e such
that
N'
= .'
fFi
7x the frame
t
3
about C, through
is the horizontal vector parallel to
the vertical plane containing the wave number vector.
frame
freu,
"
'3
is
obtained by rotating the frame
The
{fi, W,/J-
about % through an angle I first and then followed by the
rotation of the rotated frame through an angle a about the
axis 2/ which is at an angle T with ' as shown in Figure 3.
Any two pair of these three frames are related by
S~i+C5Q~a(9-1)
~.
GCsf Co i + Co sp S~,nrL
MDWM
=Cos
C
(
' 5+
G S 5 (4
sI
9
3
-64-
e2 2nI
e2
e3
e3
Figure 3.
Coordinate frame
figure
0'
.
"x
r is
''
{ (,
,
is
specified in
measured clockwise about
'
'3
is parallel to the symmetry
axis of transversely isotropic medium, and at
an angle
with the horizontal plane.
In particular, when S = 0, namely, (
lies in a horizontal
plane, (9-3) is reduced to
Sir
(-p
~
+Cos(e)'
(9-4)
The displacement vector Z in case of infinitesimal deformation
can be written as
-i +
+(1
+(C0
C05~ (1*L.
e
PS
(-L
Ca- S e~'
+pCs
(
(9-5)
Furthermore, if we assume that a symmetry axis of a homogeneous
transverse isotropic medium with five elastic constants AL,,AL,
Alu IAl
and V is arbitratily oriented, then by suitable
choice of the angular variables S and
r,
direction of the symmetry axis and
' in a horizontal plane.
ZjDW will lie in the
-66The choice of this frame outlined to the transverse isotropy
is reasonable and can simplify our calculation in deriving
the equation of motion later-on, since in the plane normal to
the symmetry axis, the frame can be arbitrarily rotated
without affecting the expression of the elastic tensor.
{ n"',*2.,
the choice of the frame
)
After
is made, the elastic tensor
can be expressed as
+~i (Al~N)('
t
(AL*
+
05
(9-6)
+
which can be decomposed into aZ=)+/
part
with Lame's constants
given by
,
c9
3
in isotropic
medium
and a perturbed part
-67-
e
't+L
+
el e
(9-7)
"+
ti + ile e3
+3 c6)
SA
where
3* '
(9-8)
+*e
+
, = A -An
= AL -Aii /A
and
77
makes
The choice of the frame
0
and hence the relation between the partial derivatives with
respect to the coordinates in
4
(e
aK
1
and frame
frame
becomes
= cO sp cs(c-r)I
p
-
e= 5A(e -r) di
32'=Slay (e.Ko-r)
a
(9-9)
(i=1,2, 3) and
where <b=)
T
=
3'
i +f=+X'A'+a
ji'
is the position vector of the medium particle.
If
motion
we apply (9-3),
-
'
(9-5) and (9-9) to the equation of
C/
3
A
A
4"
X3 I
-68we find that
A-
1 3134
(L
(9-10)
(r)
where the 3x3 matrix operator
( )
given in (2-12) and
is the same as that
is the perturbed part which
contributes to the dependence of the motion on the angle
between the direction of propagation vector and symmetry axis.
Following the same procedures as those discussed in the previous
sections, we shall calculate
and
A(V)
j
'3
when we deal with P, SV waves and surface waves
of Rayleigh type and
6
when dealing with SH waves.
For the discussion of P and Rayleigh waves, the relevant
(t',
motion in two dimensions
(Lai
E,')
will be
V3 (W)(9-11)
where-
S
Cos~
11(6
(
-~'l
5
+a-cc (e~r
h Cc3 (
-69S- -
-
5h3lpCo o(05
y (0
cg- ) a(03
cf13(0-r)
1
Cos2j Co2 3(e-0 )
+ (9Co-5
4; (go=
c32~
s2p Cosc(6sr)+<-
%3S)z(V
s(-r
- .( ( -( SA)(
(3sepCposs%30-35>pcosp
4
4'2(o-d 0)
+PSCO
(6-6
Cos
cosp
.J[Ap
+(
SA
-
6
A)
(
s1'y Co5't Co5 <-r)+
Sl$ ate-r) + (asp 4cos-)].
(A2.S).)
6)
X +coly3(s
(oS 09-r I
(e-f)
SWj Cosp GS
S ~ Cc~3 Co
,Zo1k)=
'1 2f
±6Co5a~Co$~(e~~)
r2[ 6'52 c 5 R Co S(e-) + S A 3
+
+
((
2)
2 4SA)SW3
spco(e-t)
(cost<s1 -r)
p + A 5 ('12P + >
)5
)3
) 2Jj2.
(9-12)
-70In the particular case that symmetry axis is parallel to
the free surface and at an angle
r
with f
,
equation (9-11)
is reduced to equation (4-1) with the angle e replaced by 6-r
r' F.=0
The boundary condition that normal stress
at the free surface
X
=o
can also be obtained by following
the procedure of carrying out the transformation of the tensor
components.
Among the three components of
('.
the condition
~).
0'o
=
,'
can be ignored.
,
The remaining
two are approximately
± 54 C5 2 -t 5os''(-f )H- 3
+ 2.A-5 X) S
np
Cosp
np
c(o-6)
S Co
'A
pPCo(9-) + J
.
-(
- )
,A-4SA)S 1Pcop Cos t
I
Cos2.P Cos(9-1)
C-G)
(9-13)
-71-
+o5 $2
+
2
-
S
-1
~ ((55-b
~S
-' CospcspCos
I (Oosc5-)>-i'(e-r) pcosp
+ ~ $ ,9
'i
ojo
(9-13)
+()
t
The operators
of
b ,
3I
and
is replaced by -
,
.
j'(9g)
are a linear combination
For sinusoidal disturbance,
a
which is purely imaginary in the case
of surface wave and hence we see from (9-11),
(9-12),
(9-13)
and the assumption of the displacement potentials of the form
(4-12) that the modified Rayleigh's equation will become a
polynomial with complex coefficients on the assumption that
. Q, 0.51r,
then the solution for phase velocity would be
complex in general.
It means that steady Rayleigh waves no
longer exist in homogeneous transversely isotropic medium
-72unless the symmetry axis is parallel to the free surface or
vertical, confirming the result of Synge (1957).
Let us consider a unit hemisphere H:
X-4
x"
=
(x,o)
in which any point can be specified by two angular variables
p, r,
then its differential area element would be CoSpAgdo~
Let W(p,Y)
be any weighting function that averages over H
defined in the following way:
V
W(pP-)\o
Then if
-r)
<f >=H
is any function
#(fr) w(par) Cosp4
is the average of
4A
(9-14)
over the whole hemisphere.
We can extend the average of functions to the average of any
partial differential operator
by setting
K--~ (
nAI
'f
T =1--~
ai3
(9-15)
-73then the averaged equation of motion becomes
k3t
[
( + 6 (w)~d)
(V
W)±4
+ )
(9-16)
The average of Voigt's approach is that it can simplify
our calculation.
)tr,
(S
It is easy to see that for integers fi,
C'OSt(e-r)
=O
n
if>Ap is odd positive integer.
Then equation (9-11) can be averaged to calcel out some terms
to make (9-16) become a equation of motion for a homogeneous
transversely isotropic medium with symmetry axis parallel to
the free surface.
To avoid bulky expressions, the following abbreviation
will be adopted.
+7
C 3±+
(-(1
.~i(9-17)
-74Hence equation of motion becomes
f1L, a
-,f
+rs]
Wa+d%]+4[Z&,
((3+ ~b~dILLI}
Z~c,+
44 a I
(9-18)
in which
CQ5 Co52 (8)t - A 4 Co sp
l(Aiw)+
t (S
+ 6 Cos5P $ j~1~X'e-~)
Cf)cosm46-r
Co 51 (o52(e-r) + S
f(A +
-($S),"0 COS,-(G-r)
4, +
s;
<-
C'
Ct
(Ali+2-t) +(S
%C2-
C051. (6 -
Co-(e -,)(~6A-~)1I1
-r)(2A4ox3
CO
b9
)(C54-SA)
+ 94O p3$
-)
4- 3pA
(9-19)
-75Let 4p(e) be the Rayleigh wave phase velocity.
~(e)
---
Set
and combine the theory
-
in Section 4 with (9-18) and the boundary condition at the
free surface
±
ICL3
'C
3~&
3
.
0
(9-20)
the modified Rayleigh's equation becomes
(C.-C'L') .+
4
-L'
t' --
2
LA'2)c'L'A']C
c'
( 'A'*~Z F'CA + ') - L (2 F'C' ---2.C'))
(c''A'*-
where
F'=
L'
(a
2 f''C'A'
t F'* )
=0
(9-21)
-76Let
and Cy be the P and SV wave velocity respectively, then
(9-22)
S --
L-
(9-23)
We shall consider two averaging schemes.
In one scheme,
we average the orientation of the symmetry axis over a segment
of the horizontal great circle on the hemisphere H which is
symmetric with respect to the axis parallel to
the origin.
%
through
In the other, the average is taken over a section
of H enclosed by a cone which is also symmetric with respect
to the axis parallel to 9
through the origin.
For simplicity,
we shall approximate the weighting function by two sequences of
step functions defined in the following way:
For each positive integer n and real number
a closed interval
('x)
Let
rX)
M)
O<A<1,
on the hemisphere is defined by
s,4 0.AII
Ids
0,3Ai
(9-24)
-77-
7r
\ ...........
..............
.............
...............
....
........
...........
..........
.......
.......
V.,
.....................
..........
............
..............
.... ...............
.........
..
.................
..........
........
......
.............
..
............
..............
.e....
................................
A..
...........
...
...........................
....................
.....
..
.. ...........
......
.........
..'.r
....
............
................
.....
.%
........
......
................
.................
....
..................
...............
*...................................
........
.....
.................
.
.......
...................
*.*.*.,:.*:
..........
",.*.,:
............
.. ..................
...............
X-:
.............
..........
................
.....
...........
......
..............
.......
.........
I.......
I...........
.................
V..........
* ........
..............
......
n
e2
Figure 4.
(the shaded region) on
Upper half part of
hemisphere
H =(Ku,
xaI
4. 1 +
, Ka
kIe5Nr, fj=a5ir
Boundary of APX" is defined by
-78-
E VP(
JEVnA
~.8X)
d
(/,y)d3
-E
X4 7r
2)
Figure 5.
,)
defined on closed
interval as a function of
the area below each
is equal to unity.
Jv
t,
where
rd
7
-79-
(A)
Wn
n= 0
Figure 6.
The sequence
and
r.
{W
plotted against
The volume under each
equal to unity.
"e
is
-80-
r)
P( SM
C
(9-26)
Their schematic figures are shown in Figure 4, 5 and 6.
The
significant difference between these two sequences is that
rL
gives uniform distribution of the orientation of the
symmetry axis (or crystallographic a axis) within a cone
enclosed by
D4')
on H, while
confines the uniform distribution of the symmetry axis to
the horizontal great circle within
6(p)
in
(9-25)
angular variable
and
W//*'
DI
due to the factor
which is the dirac delta function in
only.
It is obvious that both
Af*'t
are weighting functions with Dirac Delta function
in angular variables 6 and
' as their sequence limit.
It is noted that the delta function served as a weighting
function will give rise to the same equations and results
in a homogeneous transversely isotropic medium with
as the symmetry axis and each V
and
W *'9
4
will give
certain degree of uniform distribution of the orientation of
-81Table 6A.
P, SV and SH phase velocity anisotropy calculated
for Voigt scheme for the weighting function VA"-f
e (Degree)
e (P"5. )
8.8267
8.8267
9.0680
8.4991
9.3167
8.2659
9.5218
8.1424
9.6655
8.0839
9.7571
8.0566
9.8868
8.0297
Table 6B.
4.9670
4.9670
4.9873
4.9873
4.9774
4.9774
4.9500
4.9500
4.9230
4.9230
4.9028
4.9028
4.8708
4.8710
P, SV, and SH phase velocity anisotropy calculated
for Reuss scheme for the weighting function
I't
- (Degree)
0
0
1
1
2
2
3
3
4
4
5
5
14
14
4.6962
4.6962
4.7492
4.6426
4.7938
4.5965
4.8244
4.5643
4.8436
4.5439
4.8551
4.5317
4.8705
4.5152
0
90
0
90
0
90
0
90
0
90
0
90
0
90
8{6t5
8.6853
8.6853
8.9442
8.3444
9.2036
8.0950
9.4127
7.9567
9.5576
7.8873
9.6493
7.8529
9.7785
7.8156
I~ (X%)e
4.6743
4.6743
4.7340
4.6138
4.7842
4.5620
4.8187
4.5254
4.8403
4.5023
4.8531
4.4884
4.8706
4.4697
4.9362
4.9362
4.9501
4.9501
4.9433
4.9433
4.9246
4.9246
4.9061
4.9061
4.8924
4.8924
4.8704
4.8704
-82Table 6C.
P, SV, and AH phase velocity anisotropy calculated
for VRH scheme for the weighting function
8
0
90
0
90
0
90
0
90
0
90
0
90
0
90
g- ( 9 A-/)
(Degree)
8.7562
8.7562
9.0023
9.4674
9.2603
8.1809
9.4674
8.0501
9.6117
7.9862
9.7034
7.9554
9.8328
7.9234
4.6853
4.6853
4.7416
4.6282
4.7890
4.5792
4.8215
4.5449
4.8419
4.5231
4.8541
4.5101
4.8704
4.4925
4.9517
4.9517
4.9687
4.9687
4.9604
4.9604
4.9474
4.9374
4.9145
4.9145
4.8976
4.8976
4.8707
4.8707
-83-
Table 7.
--
0
0
1
1
2
2
3
3
4
4
5
5
P wave and Rayleigh wave phase velocity anisotropies
calculated for three averaging schemes by choosing
Wn (0.95) as the weight function.
----------------------------------------------------------VRH
Reuss
Voigt
00
90*
00
90*
00
90*
0*
90*
0*
90*
0*
90*
8.4940
8.4940
8.5453
8.4255
8.6034
8.3527
8.6673
8.2890
8.7355
8.2335
8.8060
8.1870
0.9972
0.9972
1.0028
0.9913
1.0086
0.9853
1.0144
0.9795
1.0201
, 0.9741
1.0254
0.9691
8.3355
8.3355
8.3926
8.2601
8.4567
8.1862
8.5264
8.1183
8.5997
8.0586
8.6750
8.0079
0.9991
0.9991
1.0048
0.9930
1.0107
0.9868
1.0167
0.9809
1.0225
0.9752
1.0279
0.9700
8.4151
8.4151
8.4693
8.3417
8.5304
8.2699
8.5971
8.2041
8.6678
8.1465
8.7407
8.0980
0.9982
0.9982
1.0038
0.9922
1.0097
0.9861
1.0156
0.9802
1.0213
0.9747
1.0267
0.9696
-84-
the symmetry axis.
The phase velocity anisotropies of the P,
S waves and Rayleigh waves calculated from the average of the
symmetry axis of the three schemes given in section 6 by
means of the weighting functions will be given in Table 6
and Table 7.
All these results will agree with the results
obtained in Section 6-8 as n goes to infinity.
Each
V
of the sequence
{
V' )5
gives consistently
large P wave velocity or higher P wave velocity anisotropies
than the observed value.
give
e%
However, the sequence
{
W
values that fit the observed data, and agree
with the observed 2% anisotropy of Rayleigh wave phase
velocity closely.
10.
Conclusion
We have presented the detailed derivation of the
modified Rayleigh's equation, and the body wave phase
velocities for a homogeneous transversely isotropic medium
in which the symmetry axis is parallel to the free surface.
The motivation for the derivation is the observed anisotropies of the Pn wave and Rayleigh wave.
We started with the 'fracture zone model', in which
the earth is considered as a lamination of fracture zone
and normal crust-mantle, approximated by an equivalent
homogeneous transversely isotropic medium in the long wave
-85-
limit.
The computed results for any reasonable values of
the rigidities of the fracture zones give much weaker
anisotropy than observed.
On the other hand, the 'Olivine'
model proposed by Francis (1969) gives much greater
anisotropy than observed.
In order to obtain agreement with observation, we
introduced a distribution of the symmetry axis of the
transverse isotropy randomly oriented about the boundary
surface of a circular cone which is symmetric with respect
to the horizontal axis normal to the ridge.
The averaging
of elastic constants over distributed orientation was
carried out by the use of a sequence of weighting function
which has the maximum in the horizontal direction
normal to the ridge axis but slowly decreasing away from this
direction.
The seismic refraction measurements show the observed
magnitude of P wave anisotropies from 3% to 8%.
Table 7
shows calculated values for the P wave and Rayleigh wave
phase velocity anisotropies for the modified Olivine
model averaged by the weighting sequence
The calculated
C
{W
N--o
values from n=2 to n=5 fall within the
range of all the measured values approximately; whereas,
the corresponding magnitudes of Rayleigh wave phase velocity
anisotropy range from 2.3% to 5.7% which is higher than
-86observed value of 2% (Forsyth, 1972).
The small discrepancy
may be attributed to the fact that the Rayleigh velocities
were measured over greater area than the Pn velocities, and
anisotropy may be diminished when averaged over a larger area.
-87References
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upper mantle under oceans, J. Geophys. Rev., 70, 3429-3439.
Backus, G.E., 1970, A geometrical picture of anisotropic elastic
tensors, Rev. Geophys. Space Phys. 8, 633-671.
Bulchwald, V.T., 1961, Rayleigh waves in transversely isotropic
media, Quart. Journ. Mech. and Applied Math., Vol. 14,
pt. 3, 293-317.
Forsyth, D., 1972, The structural evolution of an oceanic plate,
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and Computers, Banff, Alberta.
Francis, T.J.G., 1969, Generation of seismic anisotropy in
the upper mantle along the mid-oceanic ridges, Nature,
221, 162.
Grand, F.S. and G.F. West, 1965, Interpretation theory in applied
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Hess, H.H., 1964, Seismic anisotropy of the upper mantle under
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Crosson, R.S. and N.I. Christensen, 1969, Transverse isotropy
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zones, Bull. Seism. Soc. Am.,
59,
59-72.
Keen, C.E., and D.L. Barrett, 1971, A measurement of seismic
anisotropy in the northeast Pacific, Can. J. Earth Sci.,
8,
1056-1064.
-88Morris, G.B., R.W. Raitt, and G.G. Shor, 1969, Velocity
anisotropy and delay time maps of the mantle near Hawaii,
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Raitt, R.W., G.G. Shor, T.J. Francis, and G.B. Morris, 1969,
Anisotropy of the Pacific upper mantle, J. Geophys. Res.,
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Raitt, R.W., G.G. Shor, G.B. Morris, and H.K. Kirk,- 1971,
Mantle anisotropy in the Pacific Ocean, Tectonophysics,
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Smith, M.L. and F.A. Dahlen, 1973, The azimuthal dependence
of Love and Rayleigh wave propagation in a slightly
anisotropic medium, J. Geophys. Res., 78, 3321-3333.
Stoneley, R., 1949, The seismological implications of aelotropy
in continental structure, Monthly Notices Roy. Astron.
Soc., Geophys. Suppl., 5, 222-232.
Synge, J.L., 1957, Elastic waves in anisotropic media, J. Math.
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-89Acknowledgements
This work was supported by the Advanced Research
Project Agency, monitored by the Air Force Office of
Scientific Research under contract F44620-71-C-0049.